Download 1. Natural transformations Let C and D be categories, and F, G : C

1. Natural transformations
Let C and D be categories, and F, G : C → D functors. A natural transformation
T : F → G is a collection of morphisms T (x) : F (x) → G(x), one for each object x
of C, such that for each morphism f : x → y in C, the diagram
F (x)
T (x)
F (f )
F (y)
/ G(x)
G(f )
T (y)
/ G(y)
commutes.
If S : F → G is a natural transformation and T : G → H is another, then
T ◦ S : F → H is a natural transformation, defined by (T ◦ S)(x) = T (x) ◦ S(x).
There is also an identity natural transformation F → F for any functor.
A natural isomorphism T : F → G is a natural transformation which has an
inverse S : G → F . Equivalently, it is a natural transformation such that for each
object x of C, the morphism T (x) : F (x) → G(x) in D is an isomorphism.
Example 1.1. The homomorphism ∂∗ : Hn (X, A) → Hn−1 (A) is a natural transformation. It is not a natural isomorphism.
Example 1.2. In Bredon IV.3 we defined for each pointed topological space (X, x0 )
a homomorphism of abelian groups
π1 (X, x0 )/[π1 , π1 ] → H1 (X).
This gives a natural transformation between two functors from the category of
pointed topological spaces to the category of abelian groups. If we restrict to the
subcategory of path connected pointed spaces, it becomes a natural isomorphism.
2. Representable functors and Yoneda’s lemma
Let Set be the category of sets, and let C be an arbitrary category. For each
object x of C, we get a functor MorC (x, −) : C → Set. This is called the (covariant)
functor represented by x. An arbitrary (covariant) functor F : C → Set is called
representable if it is naturally isomorphic to MorC (x, −) for some x.
Let C be a category and F : C → Set a functor. Let x ∈ C, and let MorC (x, −) :
C → Set be the functor represented by x. If T : MorC (x, −) → F is a natural
transformation, it in particular gives a map T (x) : MorC (x, x) → F (x), which
we can evaluate on the identity map Idx and get an element T (x)(Idx ) ∈ F (x).
Yoneda’s lemma is the following statement.
Lemma 2.1. The association T 7→ T (x)(Idx ) gives a bijection
Nat(MorC (x, −), F ) → F (x)
from the set of all natural transformations to the set F (x).
Proof sketch. Given c ∈ F (x), we can get a natural transformation Tc : MorC (x, −) →
F by
Tc (y)
MorC (x, y) −−−→ F (y)
f 7→ F (f )(c)
This process is inverse to the process T 7→ T (x)(Idx ).